Evaluate the limit Lim (x²+x)/(sin(x))x→0
To evaluate the given limit:
lim (x² + x) / sin(x) as x approaches 0,
we can try to simplify the expression in order to apply limit properties
To evaluate the given limit:
lim (x² + x) / sin(x) as x approaches 0,
we can try to simplify the expression in order to apply limit properties.
First, let’s factor out x from the numerator:
lim x(x + 1) / sin(x) as x approaches 0.
Next, let’s consider the limit of the numerator and denominator separately:
lim x → 0 (x) * lim x → 0 (x + 1) / lim x → 0 (sin(x)).
The limit of x as x approaches 0 is simply 0:
0 * lim x → 0 (x + 1) / lim x → 0 (sin(x)).
Now let’s evaluate the limit of x + 1 as x approaches 0:
0 * (0 + 1) / lim x → 0 (sin(x)).
This simplifies to:
0 / lim x → 0 (sin(x)).
Finally, let’s evaluate the limit of sin(x) as x approaches 0:
0 / (sin(0)).
Since sin(0) is equal to 0, we have:
0 / 0.
However, this is an indeterminate form and does not provide a definite value for the limit.
In this case, we can use L’Hopital’s rule to find the limit.
Taking the derivative of the numerator and denominator and applying L’Hopital’s rule:
lim x → 0 (1 + 0) / cos(x).
Simplifying further:
lim x → 0 (1 / cos(x)).
Now, as x approaches 0, the cosine function approaches 1:
1 / cos(0) = 1 / 1 = 1.
Therefore, the limit of (x² + x) / sin(x) as x approaches 0 is equal to 1.
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